NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

These NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 Questions and Answers are prepared by our highly skilled subject experts.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.4

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Question 1.
\(\frac { { e }^{ x } }{ sinx } \)
Solution:
\(y=\frac { { e }^{ x } }{ sinx } \)
\(for\quad y=\frac { u }{ v } ,\)
\(\frac { dy }{ dx } =\frac { { e }^{ x }{ sin }x-{ e }^{ x }cosx }{ { sin }^{ 2 }x } \)
\(or\frac { dy }{ dx } =\frac { { e }^{ x }{ sin }x-{ e }^{ x }cosx }{ { sin }^{ 2 }x } ,where\quad x\neq n\pi ,x\in z \)

Question 2.
\({ e }^{ { sin }^{ -1 }x }\)
Solution:
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 1a

Question 3.
\({ e }^{ { x }^{ 3 } }\) = y
Solution:
Let y = \({ e }^{ { x }^{ 3 } }\)
Differentiating w.r.t. x,
\(\frac{d y}{d x}=\frac{d}{d x}\left(e^{x^{3}}\right)\) = \(e^{x^{3}} \cdot \frac{d}{d x}\left(x^{3}\right)\)
= \({ e }^{ { x }^{ 3 } }\).3x² = 3x²\({ e }^{ { x }^{ 3 } }\)

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Question 4.
\(sin\left( { tan }^{ -1 }{ e }^{ -x } \right)\) = y
Solution:
\(sin\left( { tan }^{ -1 }{ e }^{ -x } \right)\) = y
\(\frac { dy }{ dx } =cos\left( { tan }^{ -1 }{ e }^{ -x } \right) \frac { d }{ dx } \left( { tan }^{ -1 }{ e }^{ -x } \right) \)
\(=cos\left( { tan }^{ -1 }{ e }^{ -x } \right) \frac { 1 }{ 1+{ e }^{ -2x } } \frac { d }{ dx } \left( { e }^{ -x } \right) \)
\(=-cos\left( { tan }^{ -1 }{ e }^{ -x } \right) \frac { 1 }{ 1+{ e }^{ -2x } } .\left( { e }^{ -x } \right) \)

Question 5.
\(log(cos\quad { e }^{ x })\) = y
Solution:
\(\frac { dy }{ dx } =\frac { 1 }{ cos\quad { e }^{ x } } \left( -sin{ e }^{ x } \right) .{ e }^{ x }\quad =-tan\left( { e }^{ x } \right) \)

Question 6.
\({ e }^{ x }+{ e }^{ { x }^{ 2 } }+\)…\(+{ e }^{ { x }^{ 5 } }\) = y(say)
Solution:
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 1

Question 7.
\(\sqrt { { e }^{ \sqrt { x } } }\), x > 0
Solution:
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 2

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Question 8.
log(log x), x > 1
Solution:
y = log(log x), x > 1
Differentiating w.r.t. x,
\(\frac{d y}{d x}\) = \(\frac{1}{\log x} \cdot \frac{d}{d x}\)(log x)
= \(\frac{1}{\log x} \cdot \frac{1}{x}\) = \(\frac{1}{x \log x}\), x > 1

Question 9.
\(\frac { cosx }{ logx }\) = y(say),x>0
Solution:
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 3

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4

Question 10.
cos(log x + ex), x > 0
Solution:
NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 4

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